Ball rolling down a ramp question again (different than traditional ones)

[libin]

Hello everyone. I am new to the forum. And this is my first question.
Consider an experiment with a ball rolling down a ramp. We have to determine several factors effecting its average speed. Please CONTINUE READING BECAUSE THIS IS A DIFFERENT QUESTION THAN THE TRADITIONAL ONES. Now, the factor i was assigned to was "length of the ramp". Now, all factors (slope, angle of elevation, ball type, mass, density, height of ramp and everything else) are all controlled. My prediction was that the speed would increase as the length of the ramp increases. This is because that the acceleration down the ramp is constant. The longer the time, the higher the final speed. When i did the experiment however, i found weird answers. There was no certain pattern! I am really confused. I would like an answer or another explanation of the situation. There could be a mistake as I did the experiment in a hurry and i may have made a mistake.
Thanks in advance.

 

[what]

First question, how did you change the "length of the ramp" if both height and angle of elevation were constant? Also how did you take in account friction?

In my opinion trying to answer these questions should give you some insight on why your results don't show the pattern you were looking for.

 

[Doc Al]

Now, the factor i was assigned to was "length of the ramp". Now, all factors (slope, angle of elevation, ball type, mass, density, height of ramp and everything else) are all controlled. My prediction was that the speed would increase as the length of the ramp increases.

You thinking sounds reasonable to me. Why don't you quantify it by figuring out exactly how average speed depends on distance? (For example: If you double the distance, what happens to the average speed?)

How did you vary the distance without changing anything else? How did you calculate the average speed? What did you measure?

As far as not being careful... too late to help you with that!

 

[libin]

Ok. With the controlled factor part. In the experiment, the height was controlled by placing a book under the ramp. Now, the ramp formed is elevated at a certain angle. When i added another part to the ramp, the same book was placed.
For the average speed part. This is what i used. I recoreded down the time it took for the ball roll down the ramp. Then i measured the length of the ramp. Using the formula "S=D/T", i calculated the speed for each try. There are a total of 10 tries. So, i found the average of all my answers.
Yes, i admit i did not be careful. Because this is the first time i actually do an experiment. Before i just did only the quantitive part of physics. Could someone just predict what will happen? Thanks to "Doc AL" and "What" for your answers.

 

[Hootenanny]

Instead of considering forces, try considering energy:
At the top of the ramp the ball will have a potential energy of $mgh$, where $h$ is the vertical height of the ramp.
Now, ignoring friction the loss in potential energy will be equal to the kinetic energy:
$$\frac{1}{2}mv^2 = mg\Delta h$$
Rearranging and cancelling the masses we have:
$$v^2 = 2g\Delta h$$
Using trig to find the ramp length in terms of h gives:
$$l\sin\theta = h$$
where $\theta$ is the angle between the horizontal and the ramp.
Subsituting $h$ into the equation gives:
$$v^2 = 2g\left(h - l\sin\theta\right)$$
$$v = \sqrt{2g\left(h - l\sin\theta\right)$$

This should give you a model for you to predict what 'should' happen.

 

[Hurkyl]

And be careful when computing the average -- you can't just take the beginning and ending velocities and take their average!

E.g. the classic paradox of average velocities is that going X miles at 45 miles per hour, and X miles at 55 miles per hour does not average out to 50 miles per hour!

I would think you would need an integral -- I've not yet been able to come up with an idea for a shortcut.


(P.S. if you're interested in the fastest ramp, see Tautochrone at Mathworld or Brachistochrone at wikipedia. They are two related problems, and these are the ones with pictures. )

 

[Doc Al]

And be careful when computing the average -- you can't just take the beginning and ending velocities and take their average!

In this case you can, since the acceleration is constant.

libin used D/T to find the average speed. Also good.

 

[Doc Al]

Ok. With the controlled factor part. In the experiment, the height was controlled by placing a book under the ramp. Now, the ramp formed is elevated at a certain angle. When i added another part to the ramp, the same book was placed.

Sounds to me like you varied not just the length of the ramp, but the angle as well. If you keep the height the same, what would you expect to happen to the average speed?

For the average speed part. This is what i used. I recoreded down the time it took for the ball roll down the ramp. Then i measured the length of the ramp. Using the formula "S=D/T", i calculated the speed for each try. There are a total of 10 tries. So, i found the average of all my answers.

Sounds to me like you kept the height the same but changed the length and angle of the ramp. Perhaps you thought you were changing only the length, and keeping everything else fixed--but not so. You kept changing the angle so that the height remained fixed. Now... what would you expect to see?

 

[Hootenanny]

Sorry an error I just noticed in my last post, the equation should read:
$$v = \sqrt{2gl\sin\theta\right)$$
Where $l$ is the distance which the ball has traveled down the ramp. Apologies.

 

[Doc Al]

Unfortunately, it looks like libin kept conditions such that $h = l \sin \theta$ remained constant. I'll bet his results were just what one would expect, but not what he thought he was measuring.

 

[libin]

Thanks to all of your guy's answers. But actually this is a presentation i have to give to my grade 10 science classmates. They do not understand anything about force or energy etc. I do understand the equations concerning energy and force, but i cannot use it according to my teacher. The aim is just to show how does the length effect the speed of the ball in the simplest manner. Could someone just sum up everything here and give me an accurate hypothesis.
And for Doc Al's questions. I kept the angle the same. This is done by just adding an extra length at the begginning of the ramp. As i said, a book was placed under the ramp. I didn't change that height which is the height of the book. Now, extending the ramp doesn't change the angle for sure. It's like extending one of the rays of an angle. Now, for the height. I am confused about that as well. Then in this case, if i extend the ramp, the height has to change. If so, how can i do such an experiment.
The sad thing is that i cannot use any of the equation you guys gave me. I just need a simple explanation. Something like "as the length increases, the speed increases."
Once again, thank you guys.

 

[Doc Al]

I kept the angle the same. This is done by just adding an extra length at the begginning of the ramp. As i said, a book was placed under the ramp. I didn't change that height which is the height of the book. Now, extending the ramp doesn't change the angle for sure. It's like extending one of the rays of an angle. Now, for the height. I am confused about that as well. Then in this case, if i extend the ramp, the height has to change. If so, how can i do such an experiment.

Did I misinterpret what you described before? If you kept the angle fixed, then the height must change accordingly as the length is increased. (Did you change the height? Yes or no?) As you realize, some of the variables are mathematically linked: the height, length, and angle define a triangle. You can't change just one without affecting at least one other. You can only keep one of those three fixed at a time.

If you kept the angle fixed, while varying the length (and height) of the ramp, what did your results show?

The sad thing is that i cannot use any of the equation you guys gave me. I just need a simple explanation. Something like "as the length increases, the speed increases."

That's certainly a reasonable statement. (Assuming you kept the angle fixed.) Since the ball accelerates down the ramp, the longer it rolls the faster it goes. Thus the average speed must increase as it rolls a longer distance.

 

[libin]

Ok. The angle is fixed, but the height varied. That's what i have done. Maybe I ignore that. Now, if this is the case, is my hypothesis correct?

 

[Hootenanny]

If you varied the height as well as the length, then your hypothesis is correct.

 

[Doc Al]

Now, if this is the case, is my hypothesis correct?

Regardless of your hypothesis, what did your measurements show?

 

[Hootenanny]

Also, could you give some quantative values?

 

[libin]

OK. The reason why I did not want to show quantitative values is because they does not make sense and I think my mate who was doing the experiment with me made a huge mistake. However, I was absent doing the experiment so I can't really change what is already done.
Anyhow, there was no trend. The speed increased, then decreased, but then increased again. That's why it's so weird.
Thanks

 

[libin]

Another question. The formula that you give doesn't seem to be correct. When I substituted my data into the formula, I did not find a measurement for Sin of the angle of elevation.

 

[Doc Al]

The formula that you give doesn't seem to be correct.

I assume you are referring to Hootenanny's formula:
$$v = \sqrt{2gl\sin\theta\right)$$

That formula ignores the rotation of the ball, but no matter; it's still true that the speed is proportional to the square root of the distance.

When I substituted my data into the formula, I did not find a measurement for Sin of the angle of elevation.

Since your data shows no dependence of speed on distance, why would you think its correct? (Anyway, the angle should be one of the givens that you measured.)

 

[Doc Al]

Also realize that Hootenanny's formula is for the final speed of the ball, whereas you were finding the average speed.

 

[Hootenanny]

Yes, the ball is modeled as a particle, so the smaller the ball you use, the more accurate the formula will be. And yes, the formula allows you to calculate the instantaneous speed at any point down the length ($l$) of the ramp.

To take into account the rolling of the ball you would need the moment of inertia of the ball.

I apologise for any confusion.

 

[libin]

Ok. Then using the formula, i can still finding out the average speed can't I?

 

[Hootenanny]

$$v = \sqrt{2gl\sin\theta\right)$$

This will give you the speed at any point on the ramp. I was under the impression that you wanted final speed,not average speed.

 

[libin]

Sorry, i meant the average speed over the journey. This would only give me the final speed. But i need the average.
Thank you.

 

[Hootenanny]

The only way you could get an average velocity from that equation is inputing numbers for l, then averaging them. If you use a spreadsheet and formulae it should be quite easy.

 

[libin]

Do you mean to calculate 2 speeds then average them? I don't quite understand what do you mean.

 

[Hootenanny]

Yes, but instead of using two speeds use maybe a hundred. You can set excel up to do all the calculations for you.

 

[libin]

Thanks so much. I got it now.

 

[Doc Al]

Sorry, i meant the average speed over the journey. This would only give me the final speed. But i need the average.

Given the formula for the final speed, you should be able to figure out the expected average speed: it starts from rest and is uniformly accelerated. (See here for a review of the kinematic equations: https://www.physicsforums.com/showpost.php?p=905663&postcount=2)

I assume that for each value of ramp length you made several runs of the ball and measured the time for each run. That data should give you a good answer for the average speed of the ball for a given ramp length. (Do not combine the speeds for different lengths into some combined average--that would be gibberish.)